ARTICLE
Received 23 Oct 2012 | Accepted 27 Mar 2013 | Published 30 Apr 2013
DOI: 10.1038/ncomms2817
OPEN
Resonant tunnelling and negative differential
conductance in graphene transistors
L. Britnell1, R.V. Gorbachev2, A.K. Geim1,2, L.A. Ponomarenko1, A. Mishchenko1, M.T. Greenaway3,
T.M. Fromhold3, K.S. Novoselov1 & L. Eaves1,3
The chemical stability of graphene and other free-standing two-dimensional crystals means
that they can be stacked in different combinations to produce a new class of functional
materials, designed for specific device applications. Here we report resonant tunnelling of
Dirac fermions through a boron nitride barrier, a few atomic layers thick, sandwiched between
two graphene electrodes. The resonance occurs when the electronic spectra of the
two electrodes are aligned. The resulting negative differential conductance in the device
characteristics persists up to room temperature and is gate voltage-tuneable due to graphene’s unique Dirac-like spectrum. Although conventional resonant tunnelling devices
comprising a quantum well sandwiched between two tunnel barriers are tens of nanometres
thick, the tunnelling carriers in our devices cross only a few atomic layers, offering the
prospect of ultra-fast transit times. This feature, combined with the multi-valued form of the
device characteristics, has potential for applications in high-frequency and logic devices.
1 School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK. 2 Manchester Centre for Mesoscience and Nanotechnology,
University of Manchester, Manchester M13 9PL, UK. 3 School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK.
Correspondence and requests for materials should be addressed to L.E. (email: Laurence.Eaves@nottingham.ac.uk).
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M
ultilayer stacks of graphene and other stable, atomically
thin, two-dimensional materials such as boron nitride,
the metallic dichalcogenides and layered oxides1 offer
the prospect of creating a new class of heterostructure materials.
Devices based on these materials have been investigated in a
number of recent experimental and theoretical papers2–28. By
fine-tuning the composition of the stacks, it may be possible to
create materials with novel electronic and optical properties,
which outperform conventional semiconductors. In this article,
we use this technology to fabricate and study a resonant
tunnelling device in which carriers tunnel through a thin boron
nitride barrier layer sandwiched between two graphene electrodes
across which a bias voltage is applied. We mount the layers on the
oxidized surface of a doped silicon substrate, which we use as a
gate electrode. The additional tuning provided by the gate
electrode allows us to probe in detail the physics of resonant
tunnelling of Dirac fermions and to exploit graphene’s unique
electronic spectrum, whereby the carrier type (electron or hole)
and sheet density can be controlled by the electrostatic field
effect12,23.
Here, we observe a strong resonant peak in the current-voltage
characteristics of the device over a wide range of gate voltages
with peak-to-valley ratios (PVRs) of up to 4-to-1. The peak and
resulting negative differential conductance (NDC) persist up to
room temperature. A model for the carrier tunnelling based on a
transfer Hamiltonian approach provides a good fit to the
measured device characteristics.
Results
Device structure. The structure of our devices is shown schematically in Fig. 1a. A thick layer of hBN is placed on top of an
oxidized Si substrate and acts as an atomically-flat substrate2 on
which the active part of the device is mounted. This consists of
two graphene electrodes on either side of an atomically thin hBN
tunnel barrier. A tunnel current is generated when we apply a bias
voltage, Vb, between the bottom and top graphene electrodes; the
b
B
Vg
Top graphene
hBN
hBN substrate
1
B
T
ii
i
Vb
25
0.5
B
T
I (nA)
Bottom
graphene
Vb
Theoretical model. To make a direct comparison with the
experimental data, Fig. 1c shows theoretical simulations based on
c
50
A
Vg
Low-temperature measurements. The main body of Fig. 1b
shows the measured gate-dependent resonant tunnelling characteristics for one of our devices, device A, in which the hBN
barrier is four atomic layers thick. Unless stated otherwise, the
following discussion of the physics and modelling is focussed on
measured characteristics of this device. The most notable feature
of the data is the presence of a pronounced peak in the current,
whose amplitude and voltage position are sensitive to Vg. Beyond
the peak, there is an extended region of NDC. This high-bias
regime was not explored in earlier work on graphene-based
tunnelling transistors12. The I(Vb) curve is approximately
symmetric at VgE 20 V (black curve), but becomes
increasingly asymmetric as Vg is tuned away from this voltage.
Although the PVR is relatively small for the symmetric I(Vb)
curve, it is significantly enhanced by tuning Vg, reaching a value
ofE2 at Vg ¼ 55 V. Note also that the Vg-induced increase in
the peak height and PVR for one polarity of Vb is accompanied by
a corresponding decrease for the opposite polarity.
I (arb. units)
a
gate voltage, Vg, is applied between the doped silicon substrate
and bottom electrode. This arrangement allows us to align the
Dirac points of the two graphene electrodes while maintaining
control of their chemical potentials, as shown schematically in the
inset of Fig. 1b. Owing to the low density of states close to the
Dirac point, the bottom graphene electrode only partially screens
the charge that is induced on the Si-gate electrode by Vg; hence
the sheet carrier density in the top electrode is also influenced by
Vg. This behaviour is related to the so-called quantum
capacitance29, which is a strong effect in graphene, unlike the
case of conventional semiconductors. A relatively modest Vg is
sufficient to induce a large change in the chemical potential, mB, of
the bottom graphene layer, thus providing additional control of
the effective barrier height and transmission coefficient12,23.
0
T
ii
0
i
4
–25
–0.5
6K
Vg in 5 V steps
15 V
–20 V
–55 V
Si gate electrode
–50
–0.5
0
Vb (V)
0.5
10 K
Vg in 5 V steps
15 V
–20 V
–55 V
–1
–0.5
2
PVR
3
Silicon oxide
1
–60
0
Vb (V)
–40
–20
Vg (V)
0
0.5
Figure 1 | Graphene-BN resonant tunnelling transistor. (a) Schematic diagram of the devices. (b) Measured current-voltage characteristics of one of our
devices (device A) at 6 K. The hBN barrier is four atomic layers thick, as determined by atomic force microscopy and optical contrast13; the active area for
the flow of tunnel current is 0.3 mm2. The Vg values range, in 5 V steps, from þ 15 V (top red curve), through 20 V (black symmetric curve) to 55 V
(bottom blue curve). The inset shows schematically the relative positions of the Fermi energies (chemical potentials) of the doped Si substrate gate
electrode (represented by hatched lines) and of the two graphene layers at the peak of the I(Vb) curve in forward bias with Vg ¼ þ 15 V. (c) Theoretical
simulation of device A obtained by using the Bardeen model and including the effect of doping in both graphene electrodes. Parameters: qc 1 ¼ 12 nm;
bottom layer is n-doped at 4.4 1011 cm 2 and the top graphene p-doped at 1.0 1012 cm 2. As our top graphene layers are exposed to the environment,
we expect them to have stronger residual (B1012 cm 2) doping than the bottom layers, as often observed in partially encapsulated double-layer graphene
devices5. The top inset (i) shows the chemical potentials mT and mB in the top (T) and bottom (B) electrodes, respectively, for Vb ¼ 0 and Vg ¼ 20 V,
which corresponds to the symmetric I(Vb) shown in black; for inset (ii) Vg ¼ þ 15 V and Vb ¼ 0.3 V, which corresponds to the peak of the I(Vb) curve. The
lower inset shows the Vg dependence of the PVR ¼ Ip/Iv obtained from our simulations, where Ip,v are the currents at the peak and the valley (minimum)
beyond.
2
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the Bardeen transfer Hamiltonian approach, which includes the
effects of residual doping in the graphene electrodes on the device
characteristics. Before discussing this figure in detail, let us first
consider an idealized case in which the graphene electrodes are
undoped and their crystal lattices are aligned. If Vg ¼ Vb ¼ 0, the
Fermi energy, m, in each layer is at the Dirac point and the two
Dirac points are at the same energy, as shown in inset (a) of
Fig. 2. If Vb40, electrons accumulate in the negatively biased
electrode with an equal number of holes in the positive electrode
(inset (b) of Fig. 2). This charging of the two layers generates an
electric field, which misaligns the two Dirac points. Therefore, in
order for a carrier to tunnel from one electrode to the other, its
in-plane wavevector k must change. Such a change is forbidden
unless the tunnelling event is accompanied by a scattering process
in which the conservation of k is relaxed. Figure 2, inset (c),
shows the effect of a positive Vg on the Fermi energies in the two
graphene electrodes when Vb ¼ 0 so that the two chemical
potentials are aligned. Keeping Vg fixed, we can adjust Vb to bring
the Dirac points of the two electrodes into alignment, see Fig. 2
inset (d), thus allowing all carriers whose energies are between the
now distinct chemical potentials of the two electrodes to tunnel
resonantly.
In real devices, elastic scattering due to disorder or interaction
effects leads to relaxation of the k conservation condition, so that
the resonant feature which appears in the current-voltage curves
has a line-width and position that depend on the disorder
potential or the nature of the interaction. This type of behaviour
has been investigated in conventional III–V heterostructure
devices in which carriers tunnel resonantly through a barrier
sandwiched between two confining quantum wells, so-called ‘2Dto-2D tunnelling’30,31.
For a tunnelling electron, a disorder-induced change, q, in its
in-plane wavevector can be described in terms of a scattering
potential, VS(q). If VS(q) is weakly dependent on q, the tunnel
current increases monotonically with Vb, so no peak is observed.
However, a resonance can appear if VS(q) has a bell-shaped
0.15
B
E (eV)
1
T
a
–0.15
–0.3
0.5
0.3
K (nm–1)
B
T
d
b
I (arb. units)
b
0
B
a, c
T
c
–0.5
T
B
d
–1
–0.5
0
Vb (V)
0.5
Figure 2 | Idealized current-voltage characteristics of a resonant tunnel
transistor. Here, we assume that the chemical potentials of the two
graphene electrodes are at the Dirac points when the device is unbiased.
T ¼ 10 K. Vg ¼ 0 (black curve); Vg ¼ 10 V (red curve). Insets (a–d) show the
positions of the chemical potentials at the marked points on the I(Vb)
curves.
dependence, falling towards zero when the magnitude of q
exceeds a certain value qc, where qc 1 is the lower limit of the
modulation length of the disorder-induced scattering potential in
real space. To model our data,
we use a short-range scattering
potential, VS ðqÞ / 1= q2c þ q2 , where qc 1 ¼ 12 nm. Such a VS(q)
can arise from potential variations associated with electron-hole
puddles32, short-range disorder33,34, the Moiré pattern at
graphene-hBN interfaces10,35,36(there are three such interfaces
in our device) and/or acoustic phonon scattering. The simulations
shown in Fig. 2 are qualitatively similar to those observed for our
devices. The region of NDC corresponds approximately to
VbB:vFqc/e, the situation where the typical in-plane
wavevector change, q, approaches qc; vF is the Fermi velocity in
graphene.
Theoretical model of the measured device characteristics. We
now focus on the quantitative features in the measured I(Vb)
curves, which are not reproduced in the simulations using
undoped graphene layers. Note, first, that the measured differential conductance at Vb ¼ 0 is finite at all values of Vg, see
Fig. 1b, whereas in Fig. 2 it tends to zero at Vb ¼ 0 for the (black)
curve with Vg ¼ 0. Second, the measured I(Vb) curves are symmetric at Vg ¼ 20 V, whereas for the model calculation in Fig. 2
the curve is symmetric at Vg ¼ 0. Our model successfully reproduces these features when we take into account the effect of
unintentional, residual doping of the graphene layers. As can be
seen by comparing Fig. 1b and c, good quantitative agreement
between experiment and theory for device A is obtained when the
residual doping has the values stated in the figure caption. All the
essential features of the measured I(Vb) characteristics are then
reproduced: the Vg dependence of the linear conductance around
zero bias due to the finite density of states—see inset (i) in Fig. 1c;
the Vg-induced shift of the peak in current and the NDC region
beyond; the way in which the application of Vg enhances the peak
in I(Vb) and associated NDC in one bias direction, while
decreasing it in the other bias direction.
The strong peak in I(Vb) arises from the resonant alignment of
the energies of Dirac cones in the two electrodes and the large
number of empty states in the top electrode, which are available to
electrons tunnelling from the bottom emitter layer, as illustrated in
inset (ii) of Fig. 1c. The lower PVR in the measured I(Vb) compared
with those in our calculations may arise in part from small leakage
currents between the graphene electrodes involving other tunnelling
mechanisms or a more complex VS(q) with less sharp cutoff than
assumed in our model. Our analysis also explains the origin of the
shoulder-like feature beyond the resonant peak in reverse bias: the
shoulder occurs when Vb and Vg correspond to the alignment of the
chemical potential in the top graphene layer with the Dirac point of
the bottom layer. When this condition is satisfied, the carriers close
to the Fermi energy in one electrode tunnel into a very low density
of states in the other, so that the measured current is insensitive to
small changes of Vb, thus yielding a differential conductance close
to zero.
Resonant tunnelling and NDC at room temperature. Figure 3
showing data for another transistor, device B, illustrates the
reproducibility of the I–V characteristics for different devices and
the persistence of the resonant tunnelling effect up to room
temperature, a key property for future device applications. In this
device, a PVR of up to E4 is obtained at low T (see inset of
Fig. 3a) and the resonant peak and NDC remain clearly defined at
room temperature (Fig. 3b). The dependence of the PVR on Vg
(inset of Fig. 3a) is qualitatively similar to device A and closely
reproduces that obtained from the model (bottom right inset of
Fig. 1c).
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a
b 100
4
100
PVR
3
I (nA)
50
1
–60
0
50
2
–40
–20
Vg (V)
0
0
–50
–50
7K
Vg in 5 V steps
300 K
Vg in 5 V steps
20 V
–25 V
–60 V
–100
–0.5
0
Vb (V)
0.5
50 V
–50 V
–100
–1.5
–1
–0.5
0
0.5
Vb (V)
1
1.5
Figure 3 | Reproducibility of resonant tunnelling in double-layer graphene devices. Device B exhibits I(Vb) curves similar to those of device A. The hBN
barrier is 5 atomic layers thick; the active area for the flow of tunnel current is 0.6 mm2. (a) T ¼ 7 K; the Vg values range, in 5 V steps, from þ 20 to 60 V.
The inset shows the Vg dependence of the PVR. (b) Room-temperature I(Vb) characteristics for device B; Vg ranges from þ 50 to 50 V in 5 V steps.
Discussion
Several different mechanisms by which NDC can be generated in
graphene-based devices have been modelled theoretically, for
example, refs 3,18,27. The observation of NDC in a graphene field
effect tunnel transistor in which the current flows along the
graphene layer was also reported recently. In this device, a tunnel
barrier potential is formed in the plane by applying a gate voltage
across a silicon nitride dielectric stripe deposited on a single
graphene layer37. Our device, in which the tunnel current flows
perpendicular to the plane of the component layers, compares
very favourably with the planar device, particularly with regard to
its strong NDC and high PVR. It is also interesting to compare
our device with resonant tunnelling in double barrier devices
made from conventional semiconductor heterostructures38,39. In
those devices, the quantum well confinement provided by the two
potential barriers creates quasi-two-dimensional states through
which charged carriers can tunnel when their energy is tuned to
resonance by an applied voltage. In contrast, our devices consist
of a single-tunnel barrier, so the device speed is not limited by the
dwell time of the carriers in a central quantum well40,41. Although
the PVR of our proof-of-concept devices is significantly lower
than that achieved in state-of-the-art III–V double barrier
resonant tunnelling diodes42,43, there are good prospects of
achieving higher PVRs. In addition, as the barrier transmission
coefficient is exponentially sensitive to the number of atomic
layers in the hBN barrier13, it should be possible to achieve much
higher peak resonant current densities in devices by using, for
example, three-monolayer barriers.
We note the recent theoretical work by Feenstra et al.15,26,
which modelled carrier tunnelling in multilayer graphenedielectric-graphene tunnel diodes. That work considered how
crystal grain size and spatial misorientation of the lattices of the
two graphene electrodes can influence resonant tunnelling.
Microstructural analysis2,12,13 of the graphene and boron
nitride layers that make up our devices indicates that they have
a high level of structural perfection, so grain size effects are
unlikely to have a major influence on the behaviour of our
devices. Supplementary Note 1 discusses how we incorporate the
effects of misorientation of the graphene lattices in our model of
the device characteristics.
In summary, we have shown how to achieve gate-controlled
resonant tunnelling with high PVR and pronounced NDC in
graphene-based multilayer, ‘vertical’ transistors. These devices
could be developed further for applications in logic circuits that
4
exploit the multi-valued form of the current-voltage characteristics. The additional functionality provided by resonant tunnelling is not available in graphene ‘barristors’23 in which the current
flow arises from carriers that are thermally activated over a
Schottky potential barrier. Furthermore, one of the key factors
that limits the speed of conventional double barrier resonant
tunnelling diodes, namely the dwell time of carriers in the central
quantum well at resonance, is avoided in our proof-of-concept
device architecture. This, combined with the high mobility of the
carriers in the graphene electrodes and the atomically thin tunnel
barrier, suggests potential applications in high-speed electronics.
Methods
Device fabrication. The methods used for fabricating this type of the device are
given in refs 12 (and its supporting online material) and ref. 13.
Measurement. The measurements of the current-voltage characteristics of the
devices were made using Keithley Model 2400 voltage sources to apply the bias
voltage between the two graphene electrodes and to apply the gate voltage between
the lower graphene electrode and the doped silicon substrate, which acted as the
gate electrode. The current flowing between the electrodes was recorded by using a
Keithley 2182A nanovoltmeter to measure the voltage drop across a known
resistance. The device was mounted on a ceramic chip carrier and placed on a
sample holder inserted into an Oxford Instruments variable temperature cryostat
for the low-temperature measurements. The room temperature measurements
were made either in the same insert or in ambient conditions with the sample
mounted in a screened box. All external electrical connections were made using
standard coaxial cables via a breakout box.
Theoretical model. The theoretical model used to fit our data is described in
Supplementary Note 1.
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Acknowledgements
This work was supported by the European Research Council, Engineering and Physical
Sciences Research Council (UK), US Office of Naval Research, US Air Force Office of
Scientific Research, European Commission FP7, and the Körber Foundation.
Author contributions
A.K.G., K.S.N. and L.E. conceived and designed the experiments, which were performed
by L.B., R.V.G., L.A.P. and A.M. A.K.G., M.T.G., T.M.F., K.S.N. and L.E. analysed the
data; M.T.G., T.M.F. and L.E. wrote the Supplementary Information and developed the
model. M.T.G. wrote code and ran the model. The Letter was written by A.K.G., T.M.F.
and L.E. with contributions from all authors.
Additional information
Supplementary Information accompanies this paper at http://www.nature.com/
naturecommunications
Competing financial interests: The authors declare no competing financial interests.
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How to cite this article: Britnell, L. et al. Resonant tunnelling and negative differential
conductance in graphene transistors. Nat. Commun. 4:1794 doi: 10.1038/ncomms2817
(2013).
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NATURE COMMUNICATIONS | 4:1794 | DOI: 10.1038/ncomms2817 | www.nature.com/naturecommunications
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